SAMPLING DESIGNS OVER TIME BASED ON SPATIAL VARIABILITY OF IMAGES FOR MAPPING AND MONITORING SOIL EROSION COVER FACTOR

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1 SAMPLING DESIGNS OVER TIME BASED ON SPATIAL VARIABILITY OF IMAGES FOR MAPPING AND MONITORING SOIL EROSION COVER FACTOR Guangxing Wang W53 Turner Hall 112 S. Goodwin Ave. University of IL Urbana, IL 6181, USA Alan B. Anderson US Army Corps of Engineers, CERL P.O. Box 95 Champaign, IL, USA George Gertner W53 Turner Hall 112 S. Goodwin Ave. University of IL Urbana, IL 6181, USA ABSTRACT In the Revised Universal Soil Loss Equation, cover factor reflects the effect of ground and vegetation covers on the reduction of soil loss and it controls change of soil erosion for a specific area. Developing optimal sampling designs over time for data collection of this factor is thus critical to monitor the dynamics of soil erosion. In this study we developed an image inferred semivariogram based method to determine optimal sample size. We further explored spatial and temporal variability, and the change of sample sizes needed over time for this cover factor. In addition, we studied application of historical ground data and uncertainties to infer semivariograms by combining the Landsat thematical mapper (TM) images to determine sample sizes. Compared to the results using ground data, the semivariogram and its dynamics of the cover factor could be successfully inferred using the multi-temporal TM images. The accuracy of sample sizes obtained using the image-inferred semivariograms could meet the requirement for regional estimation, but for local estimation for mapping it was very much dependent on the quality and correlation of the images with the factor. Moreover, historical ground data should be used with great caution for sampling design. INTRODUCTION Soil erosion in the USA is usually predicted using an empirical model, that is, the Revised Universal Soil Loss Equation (RUSLE) (Renard et al., 1997). In the equation, soil loss is a function of six input factors including rainfall-runoff erosivity, soil erodibility, slope length, slope steepness, ground and vegetation cover, and support practice. The ground and vegetation cover factor simply called C factor is the rate of soil loss from an area with specified cover and reflects the effect of ground and vegetation covers on the reduction of soil loss by reducing rainfall and runoff. For a specific area, ground and vegetation covers change over time often due to disturbance from human activities. Thus, soil erosion is most sensitive to the cover factor. The cover factor is usually calculated using a set of empirical equations as functions of ground cover, canopy cover, and minimum rain drip vegetation height (Wischmeier and Smith, 1978). Ground cover includes plants, litter, bare lands, and rocks, and the canopy cover consists of trees, shrubs, and grass. The minimum rain drip vegetation height means average minimum height for rain dripping down to the ground from the lowest canopy cover layer. Often the measurements of these variables are obtained by sampling transect lines. The average ground cover, canopy cover, minimum drip height are calculated based on the samples, and corresponding sample values of the cover factor are derived using a set of empirical equations (Wang et al., 22b). The values of soil erosion at the non-sample locations are estimated using the measurements at the sampling locations by spatial interpolation ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

2 Therefore, sampling this cover factor on the ground becomes very critical for monitoring and mapping the dynamics of soil erosion. A common method is to establish permanent ground plots based on a simple random or systematic sample. However, the ground and vegetation covers and soil erosion vary spatially and temporally, and their coefficients of variation change from place to place and from time to time (Wang et al., 22a; Xiao et al., 24). Especially, the dynamics of soil erosion is complicated because human activities disturb the ground and vegetation covers and cause an increase of soil erosion, and on the other hand, the recover of ground and vegetation covers leads to decrease of soil erosion over time. Thus, a permanent sample with a fixed number of plots and obtained based on simple random or systematic sampling may not be optimal in terms of spatial distribution of plots. Also, it might not be sufficient to accurately monitoring the dynamics of soil erosion. The objective of this study is to explore spatial and temporal variation of this cover factor, and corresponding sampling designs over time. Sampling design deals with determining plot and sample size, plot shape, and spatial allocation of plots to collect ground data given a cost and desired precision of estimates. However, this study focuses on searching for optimal sample size and its change over time, given that optimal sample size is defined as the number of plots that minimizes cost given a desired precision of an estimate. There are two distinct frameworks for inference in sampling context (Thompson, 1992): design-based approach in which values of variables are regarded as fixed quantities and the selection probabilities in sampling are used in determining the expectations, and model-based approach in which values are considered to be realizations of random variables and the estimators are determined based on joint distribution of the random variables. Brus and de Gruitjter (1997) compared these two approaches and suggested a decision-tree for choosing them. If a model-based reference is applied, traditionally, a sample size for simple random sampling can be calculated based on variation of a variable given a desired precision (Curran and Williamson, 1986; De Gruijter and Ter Braak, 199). The traditional method assumes that sample data are spatially independent. It easily leads to a sample size larger than that required due to duplication of spatial information since some sample plots are inevitably close and there is often a spatial dependency of plots (Curran, 1988). Recent studies in geostatistics tend to support a regionalized variable theory, that is, sample data of random variables and between them are spatially correlated within a separation distance of data locations given a direction (Goovaerts, 1997). This spatial characteristic is also called spatial variability or spatial autocorrelation of random variables. Neglecting this feature may lead to higher costs for the sampling design and larger uncertainty in estimates (Curran, 1988). Based on kriging variance, an optimal sampling design method was developed for mapping soil properties (Atkinson, 1991; Burgess et al., 1981; McBratney, et al., 1981a and 1981b). The idea behind this method is that when kriging is used for each unobserved location or block within a study area, minimum error variance can be calculated. The error variances depends only on a semivariogram quantifying the spatial variability of a random variable, and thus, on configuration of observation plots in relation to the point or block to be estimated, and not on sample data themselves. Once the semivariogram is available, for example, by conducting a pilot survey or using existing data, the error variance for any grid spacing sampling distance between plots can be calculated before conducting the survey. By changing the grid spacing, different error variances can be obtained. If the error variance of a kriged estimate can be used as a reasonable measure of optimal sampling design, the maximum grid spacing given a desired precision can be determined. The method above assumes that a semivariogram of a mapped variable called primary variable already exists, or can be inferred by a pilot survey or from historical data. A pilot survey requires additional costs. Moreover, a reliable semivariogram is based on a good representative sample. Historical data obtained by a non-optimal sample can easily lead to a biased semivariogram and thus a non-optimal sampling design. In addition, when large changes have occurred across the landscape since past field measures were made, the semivariogram inferred using historical data might also differ greatly from the current truth, which may result in an increase in uncertainty. A potential solution for the issues given above is to infer the semivariogram of the primary variable using remotely sensed data. A remote sensing image is a representation of ground characteristics (Wang et al., 22b). If a primary variable is highly correlated with spectral values of an image, the spatial variability of this primary variable can be estimated from the semivariogram of the spectral (called secondary) variable. Almeida and Journel (1994), Goovaerts (1997) and Journel (1999) proposed Markov Models I and II. A Markov Model I requires that a cross semivariogram measuring the joint spatial variability between a primary and spectral variable can be approximated using the auto semivariogram of the primary variable, the traditional covariance between the two variables, and the traditional variance of the primary variable. A Markov Model II approximates the cross semivariogram between the primary and spectral variable using the auto semivariogram of the spectral variable, the traditional covariance between the two variables, and the traditional variance of the spectral variable. Combining both Markov Models I and II makes it possible to estimate the semivariogram of a primary variable from remotely sensed data. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

3 Moreover, the advantages of Landsat TM imagery include large and complete coverage, multi-spectral and multi-temporal characteristics. They provide the potential to capture spatial variability of a primary variable and to monitor the change of the spatial variability over time for large areas. Hence, they can be used to improve sampling designs over time for monitoring systems. When a semivariogram of a primary variable is inferred using images, in addition, the variance of the variable is still needed, but can be estimated using historical data. For monitoring of a system, however, the historical data may become significantly different from the truth as time goes on. After a certain time period, the role of the historical data may completely disappear. Thus, studies related to the uncertainty of historical data on sampling design over time are necessary. In this study, we intend to develop spectral spatial variability based method to determine optimal sample sizes over time for mapping and monitoring the ground and vegetation cover factor. Moreover, we will test application of historical ground data and their uncertainties for inferring variance needed in the sampling design for monitoring this cover factor. By comparison, we will also analyze the uncertainty of the permanent sample. We expect this study will provide some suggestions on the spatial and temporal variability based sampling designs of remotely sensed data and applications of historical ground data for monitoring of other soil erosion systems. Study Area and Data Sets The study area consisting of 87,89 ha is located at Fort Hood, Texas, where summer is long and hot, and winter is short and mild. The dominant vegetation type at the east and northeast is oak-juniper woodlands. West and south parts are savannah type, dominated by grasses with scattered motts of live oak. In the center there is a mixture of the savannah type and oak-juniper woodlands. In the entire area, ground and vegetation cover was disturbed due to human activities during the 199 s. Soil erosion has thus been a concern. A sampling design and field survey for dynamics of soil loss was conducted according to the Land Condition Trend Analysis (LCTA) plot inventory method (Tazik et al., 1992). A total of 159 permanent plots were established with a stratified random sample design based on vegetation and soil type in the summer of 1989 (Diersing et al. 1992). After that, the plots were re-measured every year until The number of plots allocated to each stratum was proportional to the percent of land area occupied by the stratum. Each plot was 1m by 6m (6m 2 ). A 1m transect line was located at the center of each plot. One hundred points were sampled along each line transect beginning at the.5m point and continuing at 1m intervals (.5m, 1.5m,, 99.5 m). Ground cover and canopy cover were observed and recorded at the 1 points. The ground cover percentage and vegetation cover percentage of each plot were calculated by dividing the total number of the covered points by the total points measured ( 1%). The minimum rain drip vegetation heights were recorded by species at.1m height intervals up to 2m, and at.5m intervals up to 8m in height. The cover factor was then derived for each field plot using the measurements of ground cover, vegetation cover, and minimum rain drip vegetation height based on the empirical models by Wischmeier and Smith (1978) A scene of six TM images at a spatial resolution of 3 m 3 m was acquired for each year from 1989 to They were respectively dated on October 16, 1989; August 16, 199; August 3, 1991; August 5, 1992; May 2, 1993; September 28, 1994; and April 8, Moreover, a set of digital orthophoto quarter quads (DOQQs) at spatial resolution of 1 m in August 1997, was acquired for this study area. Using ground GPS control points, these DOQQ images were geo-referenced to UTM, WGS84, re-sampled to 4 meters, and mosaiced together to cover the area. The mosaic was used as a base image to which the 1989 TM images were geo-referenced. The TM images of all other years were then registered to the 1989 images. A first order polynomial was applied for geo-reference and registration. The maximum root mean square error allowed was 3 pixels for geo-referencing the DOQQ images to UTM, and 1 TM pixel for registering the TM images to the DOQQ mosaic. The images were then aggregated to pixel size of 9 m by 9 m with window average of 3 3 pixels. It needs to be pointed out that we used ground and vegetation cover percentages and average rain drip vegetation heights based on the transect lines of 9m to directly approximate the corresponding values of the 9 9m 2 image pixels collocating with the transect lines. This will lead to errors. However, these errors are random for each of the collocated pixels, and the approximation is unbiased for the whole population. Thus, this approximation should not lead to change in the correlation feature of the used images with the cover factor, and also should not affect the assessment of the results. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

4 METHODS Image Transformation In addition to the six original band images for each year s scene, Nine type of data transformations were made. They include a) normalized difference vegetation index (NDVI) (TM4 + TM3) / (TM4 - TM3); b) TM5 / TM4; c) TM7 / TM4; d) (TM3 * TM5) / TM4; e) (TM3 * TM7) / TM4; f) (TM3 + TM5) / TM4; g) (TM3 + TM7) / TM4; h) (TM2 + TM3 + TM5) / TM4; and i) (TM2 + TM3 + TM7) / TM4. The purpose for the ratio transformations was to increase the correlation of the image data with the cover factor by reducing the redundant information due to high correlation between the images and by increasing variation of the image data. The correlation between the spectral variables and the cover factor was then analyzed. Spatial Correlation And Kriging For Sampling Design The cover factor shows significantly spatial autocorrelation within a range of distances given a direction (Wang et al., 22b). Let Z( u ) be a random function of the cover factor or a spectral variable defined at location u in 2- dimensional space, and suppose that in the study area, a total of n observations have been obtained with values of Z( u α ) ( α = 1,2,..., n). According to the so-called intrinsic hypothesis of stationarity (Journel and Huijbregts, 1978), an experimental semivariogram γ ( h) can be calculated as: 1 γ ( h) = ( Z( u ) Z( u + h)) 2 Nh ( ) N( h) α α 2 (1) α = 1 where h is a lag of distance given a direction, N(h) is the set of all pair-wise Euclidean distances, zu ( α ) and zu ( + α h) are data values of a variable Z at spatial locations u α and u + h α, respectively. The experimental semivariograms need to be fitted using permissible models so that the resulting functions are conditionally negative semi-definite and acceptable as variance functions. These permissible models include nugget, spherical, exponential, and Gaussian model. If a map unit V to be estimated is equal to or larger than the sample plots, the unit is called a block and the modeled semivariograms above are then applied in a block kriging estimator to derive local estimate and error variance of each block, called location estimation. The minimized error variance or their square root (standard error) for each block can be directly calculated and depends only on the semivariances by the separation distances between the observed plots, and between each plot and the block to be estimated, but not on the sample data themselves. Thus, the error variance can be used to determine the maximum distance between sampled plots given a desired precision. Furthermore, sampling is also related to spatial allocation pattern of plots across the landscape can be used. However, in this study we simply used an equilateral triangle grid configuration because of its optimality. Additionally, we also assumed that the spatial variability of the cover factor is isotropic. For other patterns of plots such as square and regular hexagonal grid and anisotropy of spatial variability, readers may refer to McBratney et al. (1981a) and Webster et al. (1989). Averaging all the estimates will result in an expected value of the study area. However, the estimates are dependent because common data are used for estimating neighboring blocks. The regional variance of the average cannot be thus calculated simply by summing variances of the local estimates. For the purpose, McBratney and Webster (1983), and Atkinson (1991) developed a Thiessen polygon method in which a region is divided into n equal polygons so such that there is one observation within each polygon S, located at the polygon center, and the observation is considered as the estimate of the polygon average. Then, the regional variance can be approximated by averaging variances of the polygons (Atkinson, 1991; McBratney and Webster, 1983). This process is called regional estimation. Changing polygon size meaning different grid spacings will lead to different regional variance. A graph showing change of the corresponding regional standard error versus grid spacing can then be developed and used to determine an optimal sample size for estimation of regional average given a desired precision. For the equations above, readers can refer to McBratney et al. (1981a) and Goovaerts (1997). ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

5 Inferring Semivariogram Of Cover Factor Using Remotely Sensed Data A cross semivariogram γ ( h ) quantifying spatial cross correlation between the cover factor and a spectral variable can be calculated as: α = 1 N( h) 1 γ ( h) = ( Zu ( α) Zu ( α + h))( Yu ( α) Yu ( α + h)) 2 Nh ( ) (2) where Yu ( α ) and Yu ( α + h) are data values of spectral variable Y at spatial locations u α and u h α +, respectively, and others are the same as in Eq. (1). According to Markov Models I (MM1) and II (MM2) (Almeida and Journel, 1994; Journel, 1999), respectively, we have: C () γ ( h) = γzz ( h) or ρ ( h) = ρ () ρzz ( h) (3) C () γ ZZ C () ( ) () ( ) ρ h = ρ ρ h ( h) = γyy ( h) or CYY () ρ ρ ρ ρ ρ YY 2 2 ZZ ( h) = () YY ( h) + [1 ()] R ( h) where γ ZZ ( h ) and ρ ( h) ZZ, γ ( h) YY and ρ ( h) YY are the semivariogram and correlogram of cover factor and spectral variable, C ZZ () and C YY () are their variances, C () and ρ () is the covariance and coefficient of correlation between them, and ρ R ( h ) is the correlogram of residual for the cover factor against the spectral variable. Combining both Markov Models I and II leads to: CZZ () γ ZZ ( h) γyy ( h) (5) C () YY Both MM1 and MM2 require the data of cover factor Z and spectral variable Y are co-located. In practice, moreover, MM1 requires the data of cover factor are collected at supports equal to or larger than those of spectral data, and the assumption is reverse for MM2. Eq. (4) can be used to estimate spatial variability of cover factor from remotely sensed data, but requires the correlogram of residual that may be obtained using historical data. When the supports for both cover factor and spectral variable are the same, the approximation Eq. (5) can be made and used to estimate the semivariogram of cover factor from remotely sensed data. When multi-temporal images are available, the optimal sample sizes given the same precisions to monitor the change of cover factor over time can be determined using block kriging variance based on the image-inferred semivariograms. However, the uncertainty is unknown. Uncertainty of Historical Data When the semivariogram of the cover factor is inferred using remotely sensed data based on Eq. (4) or Eq. (5), the residual ρr ( h ) or variance, C ZZ (), is required. This indicates that a data set in which the sample plots are independent should be obtained. Generally, a historical sample obtained by random sampling can meet this requirement. That is, the historical data can be used to estimate current variance or residual of the cover factor within a certain time period. However, the historical data can gradually becoming biased from the current truth as time increases. It is thus important to determine the time period in which the uncertainty of historical data used to estimate the current variance or residual is acceptable. In this study, a permanent sample is available and provides the possibility to explore the uncertainty of historical data in sampling design. Results Listed in Table 1 are the sample statistics over time for the ground and vegetation cover factor. The sample average decreased from 1989 to 1993, although to some extent, it fluctuated, then increased in 1994, and decreased again in The coefficients of variation for the cover factor rose from 1989 to 1994 and then slightly dropped (4) ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

6 down to The spatial distribution of the plots and the cover factor values are shown in Figure 1. The cover factor had smaller values at the east and northeast, and larger values at the west, southwest and northwest. Table 1. Sample statistics over time for the ground and vegetation cover factor Cover factor Year Mean Variance Coefficient of Variation Year 1989 Year 1991 C factor N 1 Meters Year 199 Year 1992 Year 1993 Year 1995 C factor Year 1994 Figure 1. Locations and cover factor values of 159 permanent sample plots over time. N 1 Meters The coefficients of correlation between the cover factor and spectral variables were calculated, but not shown here because of space limit. The spectral variable that had the highest correlation with the cover factor was ratio image (TM3+TM7)/TM4 for year 1989, ratio image (TM2+TM3+TM7)/TM4 for 199, ratio image TM7/TM4 for 1991, original image TM 2 for 1992, ratio image TM7/TM4 for 1993, ratio image TM3*TM7/TM4 for 1994, and ratio image TM7/TM4 for Moreover, among six original channel images the TM7 was most correlated with the cover factor for all the years except for years 1992 and TM7 was also included in all the ratio images that were most correlated with the cover factor, except for year TM7 was highly correlated with all other original images except for TM4, while TM4 had a low correlation with all other original images. Thus, the TM7 and TM4 contained most of the original spectral information, and the ratio images involved these two channels led to the highest correlation with the cover factor. The semivariogram and change of the cover factor over time are given in Figure 2. These experimental semivariograms were calculated with ground data and fitted using the same spherical model determined by nugget, structure and range parameters of spatial correlation. The nugget can be considered the noise term when the separation distance is zero, implying short distance variability and measurement error. The structure parameter accounts for structure variance. The sum of nugget and structure variances means the maximum variation, and the range parameter is defined as the distance at which the spatial variability reaches the maximum. Although slight fluctuations existed for nugget variance in 199 and 1993 and for structure variance in 199 and 1991, the nugget and structure variances of the semivariograms increased from year 1989 to 1994, and then decreased in Large increases took place from 1993 to 1994 for both of them. The changes in nugget and structure variances were consistent with the change of total variance for the cover factor in Table 1. The large nugget variances for years 1994 and 1995 implied large uncertainties of measurements of the cover factor, and large structure variances for those two years suggested significant changes of spatial variability to this factor. Most of the ranges of spatial autocorrelation varied from 4 m to 55 m, except for years 1989 and 1992, with 6479 m and 84 m, respectively. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

7 Semivariance Distance (*1m) Figure 2. Spatial variability (modeled semivariogram) of the cover factor from year 1989 to Overall, the spatial variability of the cover factor increased with time. The semivariograms can be divided into three groups: the first group for the years 1989 and 199; the second for 1991, 1992 and 1993; the third for 1994 and Significant differences among the semivariances at the same distance occur among the three groups. On one hand, this implies significant change of spatial variability and correlation of the cover factor over time, and thus, new sampling designs may be needed at the time when the spatial variability changed significantly. On the other hand, this also might suggest differences in quality of the ground data between these groups. For example, larger measurements errors might have occurred in 1994 and 1995 because of the significantly larger nugget variances. In Figure 3, the experimental and modeled semivariograms using ground and image data are compared. The relative error (1*error/(average of experimental semivariance)) of the modeled semivariograms varied from 12.51% to 17.76% for ground data-inferred semivariograms and from 14.2% to 26.43% for image-inferred semivariograms, which implied that the image data led to larger uncertainties in inferring the semivariograms than the ground data. Compared to the semivariograms using ground data, the semivariograms using TM images had smaller nugget variances and larger ranges of spatial correlation. These resulted in smaller image-inferred semivariances than those using the ground data at the same separation distances. The reason for the smaller nuggets may be because the spectral values represent the combined reflectance of ground surface, and the data aggregation from pixel size of 3 m to 9 m further smoothed the within pixel variability. The smoothing may increase the spatial correlation of spectral values and thus might be the reason for larger range parameters. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

8 (a) Year 1989 (b) Year 199 Semivariance Distance (*1m) Semivariance Distance (*1m) Exp. Semiv. Image_inferred Exp. Semiv. Image_inferred (c) Year 1991 (d) Year 1992 Semivariance Distance (*1m) Semivariance Distance (*1m) Exp. Semiv. Image_inferred Exp. Semiv. Image_inferred (e) Year 1993 (f) Year 1994 Semivariance Distance (*1m) Semivariance Distance (*1m) Exp. Semiv. Image_inferred Exp. Semiv. Image_inferred (g) Year 1995 Semivariance Distance (*1m) Exp. Semiv. Image_inferred CCCCCCCCCCCCCCCCCCCCCCCCCC Figure 3. Comparisons of experimental and modeled semivariograms obtained using ground and image data for the cover factor. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

9 (a) Year 1989 (b) Year 199 Regional block kriging standard error Regional block kriging standard error needed (c) Year 1991 (d) Year needed needed (e) Year 1993 (f) Year 1994 Regional block kriging standard error Regional block kriging standard error (g) Year Figure 4. Comparisons of block kriging standard errors versus sample size when using semivariograms derived using ground and image data for regional estimation. Compared to the semivariograms fitted based on ground data, the image-inferred semivariograms had relative errors from 6 % to 2%. Fairly small relative errors (6% - 1%) existed for years 1989, 199 and 1994 because of good quality of the images. For years 1991, 1992, 1993 and 1995, the errors of the image-inferred semivariograms were relatively large (16% to 2%). The reasons may include cloudy weather and incorrect times when the TM images were acquired. For example, the 1995 images were acquired in May and did not capture the spatial ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

10 variability of the cover factor values derived using the measurements of ground and vegetation cover, and the vegetation heights in summer and fall..2 (a) Year (b) Year 199 Local maximum block kriging standard error Local maximum block kriging standard error needed needed (c) Year needed Local maximum block kriging standard error (d) Year needed (e) Year Local maximum block kriging standard error (f) Year (g) Year Figure 5. Comparisons of block kriging standard errors versus sample size between uses of semivariograms derived using ground and image data for local estimation. ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

11 The sampling designs (block kriging standard errors versus sample size) derived based on the ground data and image-inferred semivariograms were compared in Figure 4 for regional estimation and in Figure 5 for local estimation. For regional estimation, two sampling designs using ground and image data for each year had very similar standard errors curves against sample size (Figure 4). For both of the years 1989 and 1994, two standard error curves almost overlapped. Compared to those for regional estimation in Figure 4, Figure 5 shows larger differences between two kriging standard error curves using the ground data and image for each year for local estimation. These two standard error curves against sample size were close to each other for the years 1989, 1991 and However, they were different for 199, 1993 and For 1995, the difference was particularly large. When the ground data-inferred and image-inferred semivariograms were respectively used to determine sample sizes needed for regional estimation, two sample sizes obtained given a precision were very close to each other. If the allowed error was 15%, the difference between two sample sizes obtained by the ground data-inferred and image-inferred semivariograms varied from 15% to 35% for 7 years from 1989 to 1995, and the greatest difference took place in 1991 and Figure 6 presents the change of sample size required at a relative error level of 2% from the average of 5 local maximum values over time, and comparison of sample sizes between the ground data and image-inferred semivariogram sampling designs for local estimation. The sampling by image-inferred semivariogram led to significantly smaller sample size than that by ground data-inferred semivariogram for local estimation. The differences of sample size were from 17% to 7%. There were relatively smaller differences (18% - 48%) for years 1989, 1991, 1992, 1994, and larger differences (51% - 71%) occurred for other years Year Figure 6. Sample size required at relative error level of 2% over time for local estimation needed for mapping. The comparisons of sample sizes are between the semivariograms using ground and image data. Relative error (%) of semivariogram inferred Year Semivariogram inferred using variance of current year ground data and current imagery Semivariogram inferred using variance of 1989's ground data and currect year imagery Figure 7. The relative errors (%) (error *1 / (average semivariance)) of semivariograms inferred using current images and historical ground data variances, and using current images and current ground data variances compared to the modeled semivariograms directly derived by current ground data. In Figure 7, ground data obtained in 1989 for the cover factor were used to estimate variances of this factor from 1989 to 1995, and further to infer semivariograms of this factor by combining current year images, and the obtained semivariograms were then compared to the modeled semivariograms directly derived using current year ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

12 ground data in terms of relative error. For comparison, the relative errors of the semivariograms inferred using current images and variance of current year ground data are also shown in Figure 7. This indicates that the images used to infer semivariograms in Figure 7 were the same for the same year, and the difference was due to the use of historical data at a different time to estimate the cover factor variances. Compared to those obtained by current year cover factor variances, the image-inferred semivariograms by the variances of 1989 historical data had significantly larger relative errors. The errors increased slowly from 6% to 15% from year 1989 to 199, then from 15% to 58%% from 199 to 1993, and finally from 95% and 92% from 1993 to 1994 and If a relative error of 3% is considered to be risky when using a semivariogram for sampling design, then the 1989 historical data set can be used only for 1989 and 199 sampling designs of the cover factor. Conclusions and Discussion When monitoring the dynamics of soil erosion, developing optimal sampling design for data collection of the ground and vegetation cover factor is critical to derive reliable population and local estimates. In this study, we developed an image-inferred semivariogram based method to determine optimal sample size for collecting ground data, and further explored spatial and temporal variability of this factor, and corresponding sample sizes needed to map and monitor it over time. We studied the uncertainties of the sample sizes using image-inferred semivariograms by comparing the corresponding sample sizes using ground data-inferred semivariograms. We also studied the application of historical ground data and their uncertainties for inferring semivariogram by combining TM images. This semivariogram based method for determining optimal sample size is based on spatial variability and correlation of remotely sensed data with the cover factor. The advantages of TM images, including large and complete, multi-spectral and multi-temporal cover, provide a potential to capture the spatial variability of the cover factor and to monitor its dynamics in the spatial variability over time, and thus to modify sample sizes for monitoring this factor. The block kriging that minimize error variance at each location and for the whole population provides a powerful tool to determine the maximum distance between sample plots given a precision requirement. The results showed that the semivariogram and the dynamics of cover factor were successfully inferred using multi-temporal TM images for seven years from 1989 to Compared to the semivariograms using ground data, relative errors were less than 2%. When using the image-inferred semivariograms for determining sample size needed for regional estimation, the sample size obtained was within a relative error of 35% compared to that by a ground data-inferred semivariogram. Considering the large variation of this factor (from 43% to 12% from year 1989 to 1995), the maximum error of 35% in sample size can be regarded as a small risk. Compared to ground data-inferred semivariograms, the image-inferred semivariograms led to the sample sizes with significantly large relative errors for local estimation. The main reason might be locally large variation of the cover factor in both space and time. For example, the largest values of this factor varied from.13 for year 1989 to.43 for Other reasons may include quality of TM images used, the times when they are acquired, and their correlation with the cover factor. On the other hand, this method assumes that the cover factor is highly spatially correlated with a spectral variable, and both have the same structure of spatial correlation. Otherwise, this method should be applied with a caution to determine optimal sample size for local estimation. The modeled semivariograms using ground and image data significantly increased with time, which indicates significant change in spatial autocorrelation of the cover factor took place from year to year. This corresponded to the need for increased sample sizes over time. In addition, when a historical ground data set was used to estimate variance of the cover factor and further to infer semivariogram together with current year TM images, the relative error increased rapidly with time. This tells us that we should use historical ground data for sampling design with great caution. In this study area, 159 permanent plots were established in 1989 and have been re-measured every year until The permanent sample might be sufficient for monitoring the dynamics of the cover factor from 1989 to 1991, and after that, additional sample plots may have been needed. This implies a need for monitoring through time if sample size is adequate. In addition, the nugget variances of the ground data inferred semivariograms slightly increased from 1989 to Compared to those for the years from 1989 to 1993, the nugget variances of the semivariograms for 1994 and 1995 were larger. Larger nugget variances indicate higher micro spatial variability and noise due to measurement errors. This might imply poor quality of the measurements of the cover factor in 1994 and On the other hand, larger nugget variances may lead to a need of more sample plots in both 1994 and ASPRS 26 Annual Conference Reno, Nevada May 1-5, 26

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